The Ding-Yuan Semifields

Semifields are an interesting subject of study in mathematics and have applications in engineering areas such as coding theory and cryptography. In their search for new Hadamard difference sets, Ding and Yuan discovered a family of planar functions which contains the new planar polynomial x^10 - x^6 + x^2 and the Coulter-Mathews planar polynomial x^10 + x^6 + x^2 over GF(3^m), where m is odd. This new planar polynomial x^10 - x^6 + x^2 over GF(3^m) automatically leads to a family of semifields, which are called the Ding-Yuan semifields or the Ding-Yuan variation of the Coulter-Mathews semifields (see for example the references below).

The only known finite commutative semifields before 2007 are:

• Finite field of order p^n for any positive integer n.

• Dickson semifields of order p^n for any positive integer n (1906).

• Albert's commutative twisted fields of order p^n for any positive integer n (1952).

• Knuth-Kantor semifields (1965,2003).

• Ganley semifields of order 3^{2n} for any odd n (1981).

• Cohn-Ganley semifields of order 3^{2n} (1982).

• Coulter-Mathews semifields of order 3^n for any odd n (1997).

• Penttila-Williams semifield of order 3^{10} (2001).

• Ding-Yuan semifields of order 3^n for any odd n (2006).

• Coulter-Henderson-Kosick semifield of order 3^8 (2007).

References for the naming

• L. Budaghyan and T. Helleseth, New perfect nonlinear multinomials over F_{p^{2k}} for any odd prime p, in: Proc. of SETA 2008, Lecture Notes in Computer Science, Vol. 5203, pp. 403 - 414, Springer Verlag, 2008.

• Z. Zha, G. M. Kyureghyan, X. Wang, Perfect nonlinear binomials and their semifields, Finite Fields and Their Applications, Volume 15, Issue 2, Pages 125-133 (April 2009). 

• M. Cordero and V. Jha,  Fractional dimensions in semifields of odd order, Designs, Codes and Cryptography 61 (2011), pp. 197-221.